The Local Estimand In Regression Discontinuity

What Effect Does a Sharp Regression Discontinuity Target?

A sharp regression discontinuity (RD) design assigns treatment deterministically at a threshold in a running variable. Its causal strength comes from comparing outcomes just below and just above that threshold. Its target, however, is not automatically the average treatment effect for everyone in the study population. The design targets an effect for units at the cutoff, understood as a limit from nearby running-variable values.

That distinction matters for interpretation. A treatment could be highly beneficial for units close to the threshold but have a smaller, larger, or even opposite effect elsewhere. A well-identified effect at the cutoff is therefore a precise answer to a local question; whether it answers a broader policy question depends on how effects vary and on which population is relevant.

Definition: Let $R$ be the running variable, $c$ the cutoff, and $D=1(R\geq c)$ the treatment indicator in a sharp RD design. Let $Y(1)$ and $Y(0)$ denote the potential outcomes under treatment and control. The average treatment effect at the cutoff is $$ \tau_c = E[Y(1)-Y(0)\mid R=c], $$ interpreted for a continuously distributed $R$ through the corresponding limits as $R$ approaches $c$. Under the conditions below, it is identified by the discontinuity in the conditional mean outcome: $$ \tau_c = \lim_{r\downarrow c}E[Y\mid R=r] - \lim_{r\uparrow c}E[Y\mid R=r]. $$

Here, $r\downarrow c$ denotes approaching $c$ from above and $r\uparrow c$ denotes approaching from below. Since treatment is assigned on the right side of the cutoff, the first limit is the treated-side outcome mean and the second is the control-side mean. If $R$ is continuous, the chance of observing a unit with $R=c$ exactly may be zero. The definition is still meaningful as a limit of conditional means, rather than as an average calculated from a group of units whose scores equal $c$.

Assumptions and the Local Interpretation

The interpretation requires a sharp threshold rule: units with $R\geq c$ receive treatment, and units with $R<c$ do not. It also requires observations with running-variable values arbitrarily close to the cutoff on both sides. Without nearby support on one side, the two limits cannot be compared from the design.

The central causal condition is that the conditional mean of each potential outcome varies continuously through $c$. In notation, $E[Y(1)\mid R=r]$ and $E[Y(0)\mid R=r]$ must each have well-defined limits at the cutoff, with no jump there unrelated to treatment. This is a condition about potential-outcome means; it does not require treatment effects to be the same for all units, nor does it require potential outcomes to be linear in $R$. Consistency links observed outcomes to potential outcomes: $Y=Y(D)$. The interpretation also presumes that one unit’s treatment does not alter another unit’s outcome.

Write the assignment rule.
In a sharp design, $D=1$ when $R\geq c$ and $D=0$ when $R<c$.

Identify the observed outcome on each side.
Just above the threshold, $Y=Y(1)$; just below it, $Y=Y(0)$.

Use continuity at the threshold.
Continuity means that the side-specific limits of the potential-outcome means represent their respective means at $c$.

Take the difference.
The treated-side limit minus the control-side limit is the average difference between $Y(1)$ and $Y(0)$ at the cutoff.

This argument identifies an average effect for units at the cutoff, not an effect for each individual unit there. It also makes clear why a discontinuity in outcomes is informative: under the assumptions, the expected potential outcomes would otherwise change smoothly through $c$, while treatment status changes abruptly. The later discussion of why continuity identifies the cutoff effect develops this logic further.

Worked Example: Interpreting a Discontinuity in Recovery Time

Suppose a hypothetical program is assigned to patients whose eligibility score is at least $c=50$. The outcome is days until recovery, so lower values are better. Assume the assignment rule is sharp, there are patients with scores arbitrarily close to 50 on both sides, and the mean potential recovery times are continuous at 50. Suppose the limiting mean is 16 days just above the cutoff and 12 days just below it.

The cutoff effect is calculated as the treated-side limit minus the control-side limit:

\tau_c = 16\text{ days} - 12\text{ days} = 4\text{ days}.

Thus, the treatment increases mean recovery time by 4 days at the cutoff under the stated assumptions. If a more favorable outcome is defined as fewer days to recovery, this is an unfavorable effect. It is not evidence that the program increases recovery time by 4 days for every patient, or that the effect is 4 days for patients with scores far from 50. The estimate answers a local average-effect question.

Why the Cutoff Effect May Differ From the Population Average

Define the conditional average treatment effect at running-variable value $r$ as $\tau(r)=E[Y(1)-Y(0)\mid R=r]$. The cutoff effect is $\tau(c)$, whereas the population average treatment effect is $E[Y(1)-Y(0)]$. By averaging conditional effects over the population distribution of $R$, the latter can be written as $E[\tau(R)]$. These quantities need not be equal if treatment effects vary with $R$, or if variation in other characteristics changes the conditional effect.

In particular, the cutoff effect is not an unweighted average of effects at all possible score values. A population average weights the effects by how common those values are in the target population. It can also refer to a different population altogether from the units near the RD threshold. The cutoff effect equals the population average if effects are constant across the population, but equality can also occur by coincidence; neither should be assumed from the RD result alone.

Worked Example: A Heterogeneous Effect and a Different Population Average

Consider a hypothetical population with $R$ uniformly distributed from 0 to 30 and a treatment cutoff of $c=10$. Suppose the conditional average treatment effect, measured in outcome points, is

\tau(r)=2+0.5(r-10).

At the cutoff, the local effect is

\tau_c=\tau(10)=2+0.5(10-10)=2\text{ points}.

For this uniform population, the mean running-variable value is 15. The population average treatment effect is therefore

E[\tau(R)] = 2+0.5(E[R]-10) =2+0.5(15-10) =4.5\text{ points}.

The cutoff effect is 2 points, while the population average is 4.5 points. The difference follows from the stated pattern of heterogeneity and the distribution of $R$: the population includes many units with scores above the cutoff, where the treatment effect is larger. This calculation does not mean the RD result is wrong. The two numbers describe different target quantities.

Worked Example: A Small Local Effect With a Larger Population Average

In another hypothetical setting, suppose the treatment effect is 2 outcome points for units with $|R-c|\leq 1$, including those at the cutoff, and 6 points for all other units. Assume the potential-outcome means are continuous at $c$; for example, the effect is constant in a neighborhood around $c$, so it has no jump there. Suppose 20% of the population has $|R-c|\leq 1$, and 80% lies outside that interval.

The cutoff effect is 2 points because the effect in a neighborhood of $c$ is 2. The population average is the population-weighted average of the two effects:

E[\tau(R)] = 0.20(2)+0.80(6)=0.40+4.80=5.20\text{ points}.

Here the population average is more than twice the cutoff effect. The two effect values and population shares are hypothetical, but the calculation illustrates the interpretation: a local effect can be correctly identified and still be an inadequate summary of effects for a population dominated by units farther from the cutoff.

What the Estimate Does—and Does Not—Represent

A sharp RD result is different from an average treatment effect defined over the full study population. It is also different from the average treatment effect on the treated or from an effect for a population selected by matching. Those approaches may target other groups or averages, depending on their assumptions and target-population definitions. An instrumental-variable analysis can, in some settings, target a complier average effect; a sharp RD instead uses the threshold design to target an effect at the cutoff.

The cutoff estimand must also be distinguished from an estimate produced using data in a finite neighborhood around $c$. An estimator uses observed data, and a practical analysis may use observations whose scores are not exactly at the cutoff. Such a finite-neighborhood estimate is intended to estimate the limiting discontinuity; it is not automatically the average treatment effect for everyone in that neighborhood. Its accuracy depends on the estimation method and the behavior of conditional means near the cutoff. The estimand remains the cutoff effect.

A local effect may still be directly relevant when a policy decision concerns people near the eligibility threshold. It may also inform decisions elsewhere if there is credible evidence that effects are sufficiently similar across groups, or if the result can be combined with additional information to estimate effects for a broader target population. Without such evidence, extending the cutoff estimate is an external-validity claim, not a conclusion supplied by the RD design alone.

Diagnostics, Limitations, and Common Mistakes

The assumptions describe features of the assignment process and of potential outcomes, not just properties of a fitted model. A sharp rule must in fact determine treatment at the cutoff, and there must be usable running-variable support on both sides. A visible change in treatment status that is not deterministic calls for a different design interpretation. Other concerns include a concurrent policy or measurement change at $c$, or a sorting process that makes the units just above and below the cutoff systematically different in ways that also change outcomes. These can undermine a causal interpretation even if an outcome plot shows a jump. Diagnostic tools can probe some of these concerns, but no single diagnostic proves the identifying assumptions.

The design is local by construction. If the relevant policy question concerns a population far from $c$, the RD estimate alone does not establish the answer. Extrapolating a fitted outcome relationship beyond the local region or treating a finite-bandwidth comparison as a population average does not solve this problem; it changes the question or adds assumptions.

Mistake: “The estimate is the average effect for all eligible units.” Eligibility may include scores well above $c$. The RD estimand is the average effect at the cutoff, not generally the average effect among all eligible units.
Mistake: “The treatment effect is 2 for each unit at the cutoff.” The estimand is an average conditional effect. It does not reveal every individual’s difference between two potential outcomes.
Mistake: “A local estimate is useless for policy.” It can be directly relevant for decisions about people near the threshold. Broader use requires a relevant target and evidence or assumptions supporting transport.
Mistake: “Using observations within a chosen range changes the estimand to an average effect within that range.” A finite range is part of an estimation procedure; it does not, by itself, redefine the cutoff estimand.

Key takeaway: In a sharp RD design, the causal estimand is the average treatment effect at the assignment cutoff, identified by the outcome-mean discontinuity under the design’s assumptions. It need not equal an average effect for a wider population when treatment effects vary.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions below.

A sharp RD estimate is a 3-point outcome increase at the cutoff. What population does this effect describe, and what does the number not imply about individuals?
If the treatment effect is constant at 4 points for every value of $R$, what are the cutoff effect and the population average treatment effect?
A population has two equally large groups. Their average treatment effects are 1 and 7 points. What is the population average effect? If the cutoff is in the first group, what is the cutoff effect?
Why is an estimate using observations within a finite range of the cutoff not automatically the average effect for everyone in that range?
Name one feature of the assignment or outcome process that could undermine a causal interpretation of the observed discontinuity.

Solutions. (1) It describes the average treatment effect at the cutoff, interpreted as a limit for units near that threshold. It does not say every unit has an effect of 3 points. (2) Both are 4 points, because a constant effect has the same value at the cutoff and across the population. (3) The population average is $0.5(1)+0.5(7)=4$ points; if the cutoff is in the first group, the cutoff effect is 1 point. (4) The finite range is used to estimate the limiting discontinuity; it does not turn that estimand into an average of individual effects across the range. (5) Examples include treatment not being assigned deterministically at the stated threshold, a concurrent change at the cutoff that affects outcomes, or sorting that changes the composition of units across the threshold.